Lifted Unit Propagation for Effective Grounding

A grounding of a formula $\phi$ over a given finite domain is a ground formula which is equivalent to $\phi$ on that domain. Very effective propositional solvers have made grounding-based methods for problem solving increasingly important, however for realistic problem domains and instances, the size of groundings is often problematic. A key technique in ground (e.g., SAT) solvers is unit propagation, which often significantly reduces ground formula size even before search begins. We define a “lifted” version of unit propagation which may be carried out prior to grounding, and describe integration of the resulting technique into grounding algorithms. We describe an implementation of the method in a bottom-up grounder, and an experimental study of its performance.

💡 Research Summary

The paper addresses a fundamental scalability bottleneck in declarative problem solving pipelines that rely on grounding first‑order (FO) specifications before invoking a propositional SAT or ASP solver. In the conventional workflow, a formula φ over a finite structure A is “naïvely” grounded by instantiating every quantified variable with every domain element, producing a large ground formula ψ. This ψ is then transformed into conjunctive normal form (CNF) using Tseitin encoding, after which a SAT solver applies unit propagation (UP) to simplify the CNF before search. Although UP can dramatically shrink the clause set, it is applied only after the costly grounding and CNF conversion steps, meaning that a huge amount of time and memory may already have been spent on generating and handling a massive formula that would later be eliminated.

To overcome this inefficiency, the authors propose Lifted Unit Propagation (LUP), a technique that performs the logical effect of unit propagation at the symbolic level, before any grounding takes place. LUP consists of two main phases:

1. Construction of a LUP structure LU​P(φ, A). This is a partial relational structure that records, for each sub‑formula of φ, whether it is forced to be true (⊤), forced to be false (⊥), or remains undetermined (∞) in all possible expansions of A. The construction is performed by evaluating an inductively defined set of rules derived from φ, using a semi‑naïve relational‑algebra algorithm. The result is a compact representation of the information that ordinary UP would later infer from the fully grounded CNF.

2. Grounding guided by the LUP structure. Existing top‑down (recursive) or bottom‑up (relational‑algebra) grounding algorithms are modified so that, whenever a sub‑formula is encountered, the algorithm consults LU​P(φ, A). If the LUP structure marks the sub‑formula as ⊤ or ⊥, the grounding algorithm replaces it immediately with the corresponding Boolean constant, thereby avoiding the generation of any ground atoms or clauses for that sub‑formula. In a top‑down setting this amounts to early termination of recursive calls; in a bottom‑up setting it amounts to a simple change in the base case of the relational evaluation.

The key theoretical insight is that LUP yields a complete abstraction of the unit‑propagation effect: any literal that would become unit after grounding and CNF conversion is already known at the lifted level. Consequently, the grounded formula produced after LUP is at most as large as the naïve grounding followed by UP, and often substantially smaller because the grounding algorithm itself can prune autark sub‑formulas that UP cannot eliminate.

The authors implemented LUP within Enfragmo, a bottom‑up grounder based on extended relational algebra. They also compared LUP to the previously published Grounding with Bounds (GWB) technique, which computes an incomplete symbolic approximation of UP. Experiments were conducted on a limited set of benchmarks (no arithmetic or aggregates) that naturally admit FO specifications. The empirical findings are:

• CNF formulas generated by Enfragmo + LUP are always smaller than those obtained by applying UP to the CNF of a naïve grounding, sometimes dramatically so.
• The size of LUP‑grounded CNF is also smaller than the ground formulas produced by GIDL (the top‑down grounder used in the IDP system), regardless of whether GWB is enabled.
• Grounding with LUP incurs a modest overhead compared to naïve grounding, because constructing the LUP structure requires additional computation. However, the subsequent CNF transformation is faster, and the overall SAT solving time is often reduced; in some cases it is even larger, indicating that the benefit is problem‑dependent.
• When combined with the MiniSat SAT solver, Enfragmo + LUP consistently outperforms the IDP system (GIDL + MiniSat) both with and without GWB.

The paper concludes that lifting unit propagation before grounding is a promising direction for scaling declarative problem solving to large domains and complex specifications. It also highlights several avenues for future work: extending the approach to handle arithmetic, aggregates, and other FO extensions; exploring more efficient data structures (e.g., BDDs or SAT‑based inference) for computing the LUP structure; and integrating LUP into a broader range of grounders and solvers. Overall, the contribution is a clear demonstration that reasoning about unit propagation at the symbolic level can substantially reduce grounding blow‑up, thereby improving the end‑to‑end performance of SAT/ASP‑based model expansion systems.